Up to the sign \((-1)^{q+n}\) , this is precisely the relation found in [1]} between the chiral currents \(H^{-2p+4}_m\)  from [2]} and chiral currents \(w^p_m\)  that generate a \({\rm w}_{1+\infty }\)  algebra. \(w^p_m \rightarrow (-1)^{p+m} w^p_m\)  is the composition of two automorphisms of the \({\rm w}_{1+\infty }\)  algebra: an inner automorphism \(w^p_m \rightarrow (-1)^m w^p_m\)  (given by conjugation by \(e^{i\pi w^2_0}\) ) and \(w^p_m \rightarrow (-1)^{p} w^p_m\) . Therefore, the algebra is unaffected by this relative sign and the modes \(\widehat{H}^k_n\)  obey the algebra found in [2]}.
